It seems appropriate that the Slow Session website should have a tune named Sloe…
And the Sloe is a regular favourite at the Isis Farmhouse Pub Bastard English sessions where Peter took a shine to it. According to TheSession.org, the Sloe is an old tune, with unknown origins, but there are possible records of it being played in Gloucestershire in the 1820s.
It was largely popularised by Spiers and Boden in their Sloe Gin set (at 1.46 mins) which has spread widely and is now even played in Japan (starting at 2.08 mins).
There are also several good places to hear it on Youtube:
Mandolin
Melodeon
And used to compare 13 different melodeons
Fiddle and guitar (The Sloe starts at 1.39 mins)
Sam Sweeney (fiddle) and a Lithuanian folk group
Oxford Slow Session 